What are bag-of-features in computer vision?
Introduction

Bag-of-features (BoF) (also known as bag-of-visual-words) is a method to represent the features of images (i.e. a feature extraction/generation/representation algorithm). BoF is inspired by the bag-of-words model often used in the context of NLP, hence the name. In the context of computer vision, BoF can be used for different purposes, such as content-based image retrieval (CBIR), i.e. find an image in a database that is closest to a query image.

Steps

The BoF can be divided into three different steps. To understand all the steps, consider a training dataset $D = \{x_1, \dots, x_N \}$ of $N$ training images. Then BoF proceeds as follows.

1. Feature extraction

In this first step, we extract all the raw features (i.e. keypoints and descriptors) from all images in the training dataset $D$. This can be done with SIFT, where each descriptor is a $128$-dimensional vector that represents the neighborhood of the pixels around a certain keypoint (e.g. a pixel that represents a corner of an object in the image). 

If you are not familiar with this extraction of computer vision (sometimes known as handcrafted) features, you should read the SIFT paper, which describes a feature (more precisely, keypoint and descriptor) extraction algorithm.

Note that image $x_i \in D$ may contain a different number of features (keypoints and descriptors) than image $x_j \neq x_i \in D$. As we will see in the third step, BoF produces a feature vector of size $k$ for all images, so all images will be represented by a fixed-size vector. 

Let $F= \{f_1, \dots, f_M\}$ be the set of descriptors extracted from all training images in $D$, where $M \gg N$. So, $f_i$ may be a descriptor that belongs to any of the training examples (it does not matter which training image it belongs to).

2. Codebook generation

In this step, we cluster all descriptors $F= \{f_1, \dots, f_M\}$ into $k$ clusters using k-means (or another clustering algorithm). This is sometimes known as the vector quantization (VQ) step. In fact,  the idea behind VQ is very similar to clustering and sometimes VQ is used interchangeably with clustering. 

So, after this step, we will have $k$ clusters, each of them associated with a centroid $C = \{ c_1, \dots, c_k\}$, where $C$ is the set of centroids (and $c_i \in \mathbb{R}^{128}$ in the case that SIFT descriptors have been used). These centroids represent the main features that are present in the whole training dataset $D$. In this context, they are often known as the codewords (which derives from the vector quantization literature) or visual words (hence the name bag-of-visual-words). The set of codewords $C$ is often called codebook or, equivalently, the visual vocabulary.

3. Feature vector generation

In this last step, given a new (test) image $u \not\in D$ (often called the query image in this context of CBIR), then we will represent $u$ as a $k$-dimensional vector (where $k$, if you remember, is the number of codewords) that will represent its feature vector. To do that, we need to follow the following steps.


Extract the raw features from $u$ with e.g. SIFT (as we did for the training images). Let the descriptors of $u$ be $U = \{ u_1, \dots, u_{|U|} \}$.
Create a vector $I \in \mathbb{R}^k$ of size $k$ filled with zeros, where the $i$th element of $I$ corresponds to the $i$th codeword (or cluster).
For each $u_i \in U$, find the closest codeword (or centroid) in $C$. Once you found it, increment the value at the $j$th position of $I$ (i.e., initially, from zero to one), where $j$ is the found closest codeword to the descriptor $u_i$ of the query image.

The distance between $u_i$ and any of the codewords can be computed e.g. with the Euclidean distance. Note that the descriptors of $u$ and the codewords have the same dimension because they have been computed with the same feature descriptor (e.g. SIFT).

At the end of this process, we will have a vector $I \in \mathbb{R}^k$ that represents the frequency of the codewords in the query image $u$ (akin to the term frequency in the context of the bag-of-words model), i.e. $u$'s feature vector. Equivalently, $I$ can also be viewed as a histogram of features of the query image $u$. Here's an illustrative example of such a histogram.



From this diagram, we can see that there are $11$ codewords (of course, this is an unrealistic scenario!). On the y-axis, we have the frequency of each of the codewords in a given image. We can see that the $7$th codeword is the most frequent in this particular query image.

Alternatively, rather than the codeword frequency, we can use the tf-idf. In that case, each image will be represented not by a vector that contains the frequency of the codewords but it will contain the frequency of the codewords weighted by their presence in other images. See this paper for more details (where they show how to calculate tf-idf in this context; specifically, section 4.1, p. 8 of the paper).


Conclusion

To conclude, BoF is a method to represent features of an image, which could then be used to train classifiers or generative models to solve different computer vision tasks (such as CBIR). More precisely, if you want to perform CBIR, you could compare your query's feature vector with the feature vector of every image in the database, e.g. using the cosine similarity.

The first two steps above are concerned with the creation of a visual vocabulary (or codebook), which is then used to create the feature vector of a new test (or query) image.

A side note

As a side note, the term bag is used because the (relative) order of the features in the image is lost during this feature extraction process, and this can actually be a disadvantage.

Further reading

For more info, I suggest that you read the following papers 


Video Google: A Text Retrieval Approach to Object Matching in Videos (2003) by Sivic and Zisserman
A Bayesian Hierarchical Model for Learning Natural Scene Categories (2005) by Fei-Fei and Perona
Introduction to the Bag of Features Paradigm for Image Classification and Retrieval (2011) by O'Hara and Draper
Bag-of-Words Representation in Image Annotation: A Review (2012) by Tsai